DOCSLIB.ORG

The Class Equation and the Commutativity Degree for Complete Hypergroups

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
The Class Equation and the Commutativity Degree for Complete Hypergroups mathematics Article The Class Equation and the Commutativity Degree for Complete Hypergroups Andromeda Cristina Sonea 1,* and Irina Cristea 2,* 1 Faculty of Mathematics, Al. I. Cuza University, Bd. Carol I 13, 700506 Iasi, Romania 2 Centre for Information Technologies and Applied Mathematics, University of Nova Gorica, Vipavska Cesta 13, 5000 Nova Gorica, Slovenia * Correspondence: [email protected] (A.C.S.); [email protected] (I.C.); Tel.: +386-0533-15-395 (I.C.) Received: 19 November 2020; Accepted: 17 December 2020; Published: 21 December 2020 Abstract: The aim of this paper is to extend, from group theory to hypergroup theory, the class equation and the concept of commutativity degree. Both of them are studied in depth for complete hypergroups because we want to stress the similarities and the differences with respect to group theory, and the representation theorem of complete hypergroups helps us in this direction. We also find conditions under which the commutativity degree can be expressed by using the class equation. Keywords: complete hypergroup; commutativity degree; conjugacy class; class equation 1. Introduction In a non-abelian group—or, more generally, in a non-abelian algebraic structure—it makes sense to compute the probability that two elements commute. This problem was first addressed by Miller [1] in 1944, when he studied the relative number of non-invariant operators in a group. Later on, Erdös and Turan [2] introduced the concept of commutativity degree as the probability that an arbitrary element x in a finite group G commutes with another arbitrary element y in G, that is, d(G) = jf(x, y) 2 G × G j xy = yxgj /(jGj2). After that, many studies have been developed to determine some bounds for this degree. For example, Gustafson [3] and MacHale [4] proved 5 independently in 1974 that for a non-abelian finite group G, the commutativity degree d(G) ≤ 8 . Other upper bounds for commutativity degree in terms of centralizers have been obtained for the dihedral group Dn by Omer et al. [5]. A classification of the groups for which the commutativity degree is above 11/32 was given in 1979 by Rusin [6], while in 2001, Lescot [7] classified the finite h 1 i groups with d(G) 2 2 , 1 , just to recall only some of the studies on this topic. The notion of commutativity degree was generalized in different ways later on. For example, the probability that an element of a given subgroup of a finite group commutes with an element of the group was studied in [8], and the probability that the commutator of a pair of elements of a finite group equals a certain given number was investigated in [9]. The notion of the subgroup commutativity degree of finite groups was proposed by T˘arn˘auceanu[10] as the probability that two subgroups of a given group commute, that is, the probability that the product of two subgroups is again a subgroup. On the other hand, the concept of commutativity degree can also be studied in other algebraic structures, such as in hypergroups. These are a natural generalization of groups, where the operation is substituted by a hyperoperation, i.e., a function defined on the cartesian product of the support set H with values in P ∗ (H), the family of all non-empty subsets of H. Thus, the result of the combination of two elements from the support set is not just an element, as in the classical algebraic structures, but a nonempty subset of the initial set. The hypergroups were introduced by F. Marty in 1934 with their theoretical meaning: The quotient of a group by any of its normal subgroups is again a Mathematics 2020, 8, 2253; doi:10.3390/math8122253 www.mdpi.com/journal/mathematics Mathematics 2020, 8, 2253 2 of 15 group; while considering a non-normal subgroup, the quotient may be endowed with a hypergroup structure. The recently published paper by Massouros [11] presents the course of development from the hypergroup, as it was initially defined by F. Marty, to the hypergroups endowed with more axioms, and this is used nowadays. The theory of algebraic hypercompositional structures, particularly the hypergroups and hyperrings, becomes then a flourishing area of Modern Algebra, and in the last years, more and more studies in algebraic geometry [12–14], number theory [15], scheme theory [16], tropical geometry [17], and theory of matroids [18] have highlighted the important role of these structures. In this paper, in the context of complete hypergroups, we address two correlated problems: the class equation and the commutativity degree. We consider this particular type of hypergroup because, by using their representation theorem, i.e., Theorem1, we notice that there exists a strong relationship between groups and complete hypergroups. Any complete hypergroup can be obtained using a group, and vice-versa, any group can be viewed as a complete hypergroup. The structure of our study is as follows. First, in Section2, we fix the notation and terminology, and we recall the basic definitions related with the class equation and commutativity degree in group theory, as well as those related to complete hypergroups, which will be used in the following. The class equation for complete hypergroups is established in Section3, while in Section4, we study the commutativity degree for these hypergroups. In addition, we explain that, in order to compute the number of the elements that commute in a finite complete hypergroup, it is necessary to compute the number of the elements that commute in a group, so the commutativity degree of a group influences the commutativity degree of a complete hypergroup (see Theorem4). This connection is also clearly described with several examples that show that the commutativity degree of a complete hypergroup depends on the decomposition of the hypergroup. Thus, the finite complete hypergroups of a certain cardinality and constructed with the same finite group can have different commutativity degrees. Finally, we calculate the commutativity degree using the conjugacy classes (see Theorem5)—the same notion used for determining the class equation. The last section contains the conclusions of our study and some open problems related to it. 2. Preliminaries In this section, we first recall the basic notions and results about the class equation and commutativity degree in group theory. Most of them are gathered in the M.A. Thesis of Ref. [19]. Secondly, we briefly present a short review of complete hypergroups, since in the next sections, we will determine the class equation and commutativity degree for such hypergroups. For more details regarding the theory of hypergroups, the reader is refereed to the fundamental books of Refs. [20–22]. 2.1. Class Equation and Commutativity Degree for Groups Let (G, ·) be a group and Z(G) = fa 2 G j ab = ba, 8b 2 Gg be the center of the group G. Definition 1. Ref. [19] We say that two elements a and b are conjugated in G, denoted here by a ∼G b, if there exists g 2 G such that a = g−1bg. The relation ∼G is an equivalence relation, and the equivalence class of each element a 2 G, i.e., [a] = fb 2 G j 9g 2 G : b = gag−1g, is called the conjugacy class of a. For a finite group G, denote by Sk(G) k(G) the number of distinct conjugacy classes of G, i.e., G = i=1 [xi]. We may recall now the famous class equation in group theory: k(G) jGj = jZ(G)j + ∑ j[xi]j. (1) i=jZ(G)j+1 Mathematics 2020, 8, 2253 3 of 15 The class equation can be related to another important notion in group theory, one of commutativity degree, which represents the probability that two elements of a group commute [3]. It is defined as follows: jf(a, b) 2 G2 j a · b = b · agj d(G) = . (2) jGj2 Let us enumerate some properties of the commutativity degree d(G) [19]: 1.0 < d(G) ≤ 1 for any finite group G; 2. d(G) = 1 if and only if G is an abelian group; 3. d(G) ≤ d(H)d(G/H) for any finite group G and H, a normal subgroup of G. 4. The function d(G) is multiplicative, i.e., d(G1 × G2) = d(G1)d(G2), for any two finite groups G1 and G2. ( ) = k(G) 5. d G jGj . 2.2. Complete Hypergroups Let (H, ◦) be a hypergroupoid, that is, a nonempty set H endowed with a multi-valued operation ◦ : H × H −! P ∗(H) (also called hyperoperation), where by P ∗(H), we denote the family of nonempty subsets of H. For any nonempty subsets A, B ⊆ H, we denote [ A ◦ B = (a ◦ b), a2A,b2B and thus, if B contains only one element b, we simply write A ◦ b instead of A ◦ fbg. Now, the reproduction axiom and associativity have sense and they appear in the definition of the following hypercompositional structures: (i) A semihypergroup is an associative hypergroupoid (H, ◦), i.e., for all a, b, c 2 H, (a ◦ b) ◦ c = a ◦ (b ◦ c). (ii) A quasihypergroup is a hypergroupoid (H, ◦) that satisfies the reproduction axiom: for all a 2 H, H ◦ a = a ◦ H = H. (iii)A hypergroup is a semihypergroup that is also a quasihypergroup. In any group, there exists only one identity, and each element has a unique inverse. This property does not hold anymore in hypergroups, in the sense that there may exist (or not) more identities, and each element may have more inverses, or none. An element e 2 H is called an identity or unit if, for all a 2 H, a 2 a ◦ e \ e ◦ a.
Load more

Recommended publications
  • Mathematics of the Rubik's Cube
    Mathematics of the Rubik's cube Associate Professor W. D. Joyner Spring Semester, 1996{7 2 \By and large it is uniformly true that in mathematics that there is a time lapse between a mathematical discovery and the moment it becomes useful; and that this lapse can be anything from 30 to 100 years, in some cases even more; and that the whole system seems to function without any direction, without any reference to usefulness, and without any desire to do things which are useful." John von Neumann COLLECTED WORKS, VI, p. 489 For more mathematical quotes, see the first page of each chapter below, [M], [S] or the www page at http://math.furman.edu/~mwoodard/mquot. html 3 \There are some things which cannot be learned quickly, and time, which is all we have, must be paid heavily for their acquiring. They are the very simplest things, and because it takes a man's life to know them the little new that each man gets from life is very costly and the only heritage he has to leave." Ernest Hemingway (From A. E. Hotchner, PAPA HEMMINGWAY, Random House, NY, 1966) 4 Contents 0 Introduction 13 1 Logic and sets 15 1.1 Logic................................ 15 1.1.1 Expressing an everyday sentence symbolically..... 18 1.2 Sets................................ 19 2 Functions, matrices, relations and counting 23 2.1 Functions............................. 23 2.2 Functions on vectors....................... 28 2.2.1 History........................... 28 2.2.2 3 × 3 matrices....................... 29 2.2.3 Matrix multiplication, inverses.............. 30 2.2.4 Muliplication and inverses...............
    [Show full text]
  • Chapter 3: Transformations Groups, Orbits, and Spaces of Orbits
    Preprint typeset in JHEP style - HYPER VERSION Chapter 3: Transformations Groups, Orbits, And Spaces Of Orbits Gregory W. Moore Abstract: This chapter focuses on of group actions on spaces, group orbits, and spaces of orbits. Then we discuss mathematical symmetric objects of various kinds. May 3, 2019 -TOC- Contents 1. Introduction 2 2. Definitions and the stabilizer-orbit theorem 2 2.0.1 The stabilizer-orbit theorem 6 2.1 First examples 7 2.1.1 The Case Of 1 + 1 Dimensions 11 3. Action of a topological group on a topological space 14 3.1 Left and right group actions of G on itself 19 4. Spaces of orbits 20 4.1 Simple examples 21 4.2 Fundamental domains 22 4.3 Algebras and double cosets 28 4.4 Orbifolds 28 4.5 Examples of quotients which are not manifolds 29 4.6 When is the quotient of a manifold by an equivalence relation another man- ifold? 33 5. Isometry groups 34 6. Symmetries of regular objects 36 6.1 Symmetries of polygons in the plane 39 3 6.2 Symmetry groups of some regular solids in R 42 6.3 The symmetry group of a baseball 43 7. The symmetries of the platonic solids 44 7.1 The cube (\hexahedron") and octahedron 45 7.2 Tetrahedron 47 7.3 The icosahedron 48 7.4 No more regular polyhedra 50 7.5 Remarks on the platonic solids 50 7.5.1 Mathematics 51 7.5.2 History of Physics 51 7.5.3 Molecular physics 51 7.5.4 Condensed Matter Physics 52 7.5.5 Mathematical Physics 52 7.5.6 Biology 52 7.5.7 Human culture: Architecture, art, music and sports 53 7.6 Regular polytopes in higher dimensions 53 { 1 { 8.
    [Show full text]
  • Finiteness of $ Z $-Classes in Reductive Groups
    FINITENESS OF z-CLASSES IN REDUCTIVE GROUPS SHRIPAD M. GARGE. AND ANUPAM SINGH. Abstract. Let k be a perfect field such that for every n there are only finitely many field extensions, up to isomorphism, of k of degree n. If G is a reductive algebraic group defined over k, whose characteristic is very good for G, then we prove that G(k) has only finitely many z-classes. For each perfect field k which does not have the above finiteness property we show that there exist groups G over k such that G(k) has infinitely many z-classes. 1. Introduction Let G be a group. A G-set is a non-empty set X admitting an action of the group G. Two G-sets X and Y are called G-isomorphic if there is a bijection φ : X → Y which commutes with the G-actions on X and Y . If the G-sets X and Y are transitive, say X = Gx and Y = Gy for some x ∈ X and y ∈ Y , then X and Y are G-isomorphic if and only if the stabilizers Gx and Gy of x and y, respectively, in G are conjugate as subgroups of G. One of the most important group actions is the conjugation action of a group G on itself. It partitions the group G into its conjugacy classes which are orbits under this action. However, from the point of view of G-action, it is more natural to partition the group G into sets consisting of conjugacy classes which are G- arXiv:2001.06359v1 [math.GR] 17 Jan 2020 isomorphic to each other.
    [Show full text]
  • Math 3230 Abstract Algebra I Sec 3.7: Conjugacy Classes
    Math 3230 Abstract Algebra I Sec 3.7: Conjugacy classes Slides created by M. Macauley, Clemson (Modified by E. Gunawan, UConn) http://egunawan.github.io/algebra Abstract Algebra I Sec 3.7 Conjugacy classes Abstract Algebra I 1 / 13 Conjugation Recall that for H ≤ G, the conjugate subgroup of H by a fixed g 2 G is gHg −1 = fghg −1 j h 2 Hg : Additionally, H is normal iff gHg −1 = H for all g 2 G. We can also fix the element we are conjugating. Given x 2 G, we may ask: \which elements can be written as gxg −1 for some g 2 G?" The set of all such elements in G is called the conjugacy class of x, denoted clG (x). Formally, this is the set −1 clG (x) = fgxg j g 2 Gg : Remarks −1 In any group, clG (e) = feg, because geg = e for any g 2 G. −1 If x and g commute, then gxg = x. Thus, when computing clG (x), we only need to check gxg −1 for those g 2 G that do not commute with x. Moreover, clG (x) = fxg iff x commutes with everything in G. (Why?) Sec 3.7 Conjugacy classes Abstract Algebra I 2 / 13 Conjugacy classes Proposition 1 Conjugacy is an equivalence relation. Proof Reflexive: x = exe−1. Symmetric: x = gyg −1 ) y = g −1xg. −1 −1 −1 Transitive: x = gyg and y = hzh ) x = (gh)z(gh) . Since conjugacy is an equivalence relation, it partitions the group G into equivalence classes (conjugacy classes). Let's compute the conjugacy classes in D4.
    [Show full text]
  • A STUDY on the ALGEBRAIC STRUCTURE of SL 2(Zpz)
    A STUDY ON THE ALGEBRAIC STRUCTURE OF SL2 Z pZ ( ~ ) A Thesis Presented to The Honors Tutorial College Ohio University In Partial Fulfillment of the Requirements for Graduation from the Honors Tutorial College with the degree of Bachelor of Science in Mathematics by Evan North April 2015 Contents 1 Introduction 1 2 Background 5 2.1 Group Theory . 5 2.2 Linear Algebra . 14 2.3 Matrix Group SL2 R Over a Ring . 22 ( ) 3 Conjugacy Classes of Matrix Groups 26 3.1 Order of the Matrix Groups . 26 3.2 Conjugacy Classes of GL2 Fp ....................... 28 3.2.1 Linear Case . .( . .) . 29 3.2.2 First Quadratic Case . 29 3.2.3 Second Quadratic Case . 30 3.2.4 Third Quadratic Case . 31 3.2.5 Classes in SL2 Fp ......................... 33 3.3 Splitting of Classes of(SL)2 Fp ....................... 35 3.4 Results of SL2 Fp ..............................( ) 40 ( ) 2 4 Toward Lifting to SL2 Z p Z 41 4.1 Reduction mod p ...............................( ~ ) 42 4.2 Exploring the Kernel . 43 i 4.3 Generalizing to SL2 Z p Z ........................ 46 ( ~ ) 5 Closing Remarks 48 5.1 Future Work . 48 5.2 Conclusion . 48 1 Introduction Symmetries are one of the most widely-known examples of pure mathematics. Symmetry is when an object can be rotated, flipped, or otherwise transformed in such a way that its appearance remains the same. Basic geometric figures should create familiar examples, take for instance the triangle. Figure 1: The symmetries of a triangle: 3 reflections, 2 rotations. The red lines represent the reflection symmetries, where the trianlge is flipped over, while the arrows represent the rotational symmetry of the triangle.
    [Show full text]
  • MAT301H1S Lec5101 Burbulla
    Chapter 11: Fundamental Theorem of Abelian Groups Chapter 24: Conjugacy Classes MAT301H1S Lec5101 Burbulla Week 10 Lecture Notes Winter 2020 Week 10 Lecture Notes MAT301H1S Lec5101 Burbulla Chapter 11: Fundamental Theorem of Abelian Groups Chapter 24: Conjugacy Classes Chapter 11: Fundamental Theorem of Abelian Groups Chapter 24: Conjugacy Classes Week 10 Lecture Notes MAT301H1S Lec5101 Burbulla Chapter 11: Fundamental Theorem of Abelian Groups Chapter 24: Conjugacy Classes Properties of External Direct Products Let G1; G2;:::; Gn; G; H be finite groups. Recall (from Chapter 8): 1. The order of (g1; g2;:::; gn) 2 G1 ⊕ G2 ⊕ · · · ⊕ Gn is lcm(jg1j; jg2j;:::; jgnj): 2. Let G and H be cyclic groups. Then G ⊕ H is cyclic if and only if jGj and jHj are relatively prime. 3. The external direct product G1 ⊕ G2 ⊕ · · · ⊕ Gn of a finite number of finite cyclic groups is cyclic if and only if jGi j and jGj j are relatively prime, for each pair i 6= j: 4. Let m = n1n2 ··· nk : Then Zm ≈ Zn1 ⊕ Zn2 ⊕ · · · ⊕ Znk if and only if ni and nj are relatively prime for each pair i 6= j: Week 10 Lecture Notes MAT301H1S Lec5101 Burbulla Chapter 11: Fundamental Theorem of Abelian Groups Chapter 24: Conjugacy Classes Example 1 (from Chapter 8) Z30 ≈ Z2 ⊕ Z15 ≈ Z2 ⊕ Z3 ⊕ Z5: Consequently, Z2 ⊕ Z30 ≈ Z2 ⊕ Z2 ⊕ Z3 ⊕ Z5: Similarly, Z30 ≈ Z3 ⊕ Z10 and consequently Z2 ⊕ Z30 ≈ Z2 ⊕ Z3 ⊕ Z10 ≈ Z6 ⊕ Z10: This shows that Z2 ⊕ Z30 ≈ Z6 ⊕ Z10: But Z2 ⊕ Z30 is not isomorphic to Z60; one has an element of order 60, but the other one doesn't.
    [Show full text]
  • Conjugacy Class As a Transversal in a Finite Group
    Journal of Algebra 239, 365᎐390Ž. 2001 doi:10.1006rjabr.2000.8684, available online at http:rrwww.idealibrary.com on A Conjugacy Class as a Transversal in a Finite Group Alexander Stein Uni¨ersitat¨ Kiel, 24118 Kiel, Germany View metadata, citation and similar papers at core.ac.uk brought to you by CORE Communicated by Gernot Stroth provided by Elsevier - Publisher Connector Received September 1, 2000 The object of this paper is the following THEOREM A. Let G be a finite group and let g g G. If g G is a trans¨ersal to some H F G, then² gG : is sol¨able. Remarks.1IfŽ. g G is a transversal, the coset H contains exactly one aGg F a <<<s G < s g g . Therefore H CgGŽ .. On the other hand G : H g < a < s a aGg G : CgGGŽ . ; therefore H CgŽ . for some g g . Ž.2Ifg G is a transversal, then it is both a left and a right transversal as g eh s hg eh if g, e g G, h g H and therefore G s g GH s Hg G. As an abbreviation we define: DEFINITION. Let G be a finite group and let g g G such that G s G gCGŽ. g. Call G a CCCP-group, where CCCP stands for conjugacy class centralizer product. The idea to study these groups came from a paper by Fischerwx 10 . For details of his work and the relation to Theorem A we refer to Section 1, but we will give here a short summary: Fischer defines a so called ‘‘distributive quasigroup’’ Q and a certain finite group G s GQŽ.F Aut Ž.Q .
    [Show full text]
  • Theorem: the Order of a Conjugacy Class of Some Element Is Equal to the Index of the Centralizer of That Element
    Theorem: The order of a conjugacy class of some element is equal to the index of the centralizer of that element. In symbols we say: jCl(a)j = [G : CG(a)] Proof: Since [G : CG(a)] is the number of left cosets of CG(a), we want to define a 1-1, onto map between elements in Cl(a) and left cosets of CG(a). Let H = CG(a). Since the elements of Cl(a) are conjugates of a, we define f by: f : xax−1 7−! xH This is 1-1 since if f(x) = f(y) then xH = yH and thus xh = y. But then x−1y 2 H so that x−1y commutes with a and therefore: x−1ya = ax−1y. From that we conclude that yay−1 = xax−1 proving 1-1. As for onto, just note that for any x 2 G, xax−1 is in Cl(a). However we also need to show the map is \well-defined”. I.e. we need to show that if xax−1 and yay−1 are two descriptions of the same element in Cl(a), then f(x) = f(y). The calculation which shows this is just the reversal of the steps above. To be precise: xax−1 = yay−1 ) ax−1y = x−1ya ) x−1y commutes with a ) x−1y 2 H ) xH = yH (Make sure you understand this last step.) QED Note that in a finite group this implies that jCl(a)j divides jGj. This observation lets us prove the following result: Theorem: If jGj = pm for p a positive prime, then Z(G) (the set of all elements that commute with all elements in the group) is non-trivial.
    [Show full text]
  • 2.2 the Centre, Centralizers and Conjugacy
    2.2 The centre, centralizers and conjugacy Definition 2.2.1. LetG be a group with operation�. The centre ofG, denoted byZ(G) is the subset ofG consisting of all those elements that commute with every element ofG, i.e. Z(G) ={x G:x�g=g�x for allg G}. ∈ ∈ Note that the centre ofG is equal toG if and only ifG is abelian. Example 2.2.2. What is the centre ofGL(n,Q), the group ofn n invertible matrices with rational × entries (under matrix multiplication)? Solution (Summary): Suppose thatA belongs to the centre of GL(n,Q) (soA is an n invertible × matrix). Fori,j in the range 1, . ,n withi=j, letE denote the matrix that has 1 in the(i,j) � ij position and zeros in all other positions. ThenI n +E ij GL(n,Q) and ∈ A(I +E ) = (I +E )A= A+AE =A+E A= AE =E A. n ij n ij ⇒ ij ij ⇒ ij ij NowAE ij has Columni ofA as itsjth column and is otherwise full of zeros, whileE ijA has Row j ofA as itsith row and is otherwise full of zeros. In order for these two matrices to be equal for alli andj, it must be that the off-diagonal entries ofA are all zero and that the entries on the main diagonal are all equal to each other. ThusA= aI n, for somea Q,a= 0. On the other hand it is ∈ � easily checked that any matrix of the form aIn wherea Q does commute with all other matrices.
    [Show full text]
  • Conjugacy Growth and Hyperbolicity
    Conjugacy growth and hyperbolicity Laura Ciobanu University of Neuch^atel,Switzerland Webinar, December 3, 2015 1 / 48 Summary 1. Conjugacy growth in groups 2. Conjugacy growth series in groups 3. Rivin's conjecture for hyperbolic groups 4. Conjugacy representatives in acylindrically hyperbolic groups 2 / 48 and by jgjc the conjugacy length of [g], where jgjc is the length of the shortest h 2 [g], with respect to X . I The conjugacy growth function is then σG;X (n) := ]f[g] 2 G j jgjc = ng: Counting conjugacy classes Let G be a group with finite generating set X . I Denote by [g] the conjugacy class of g 2 G 3 / 48 I The conjugacy growth function is then σG;X (n) := ]f[g] 2 G j jgjc = ng: Counting conjugacy classes Let G be a group with finite generating set X . I Denote by [g] the conjugacy class of g 2 G and by jgjc the conjugacy length of [g], where jgjc is the length of the shortest h 2 [g], with respect to X . 3 / 48 Counting conjugacy classes Let G be a group with finite generating set X . I Denote by [g] the conjugacy class of g 2 G and by jgjc the conjugacy length of [g], where jgjc is the length of the shortest h 2 [g], with respect to X . I The conjugacy growth function is then σG;X (n) := ]f[g] 2 G j jgjc = ng: 3 / 48 I Conjecture (Guba-Sapir): most (excluding the Osin or Ivanov type `monsters') groups of standard exponential growth should have exponential conjugacy growth.
    [Show full text]
  • A Note on Conjugacy Classes of Finite Groups
    Proc. Indian Acad. Sci. (Math. Sci.) Vol. 124, No. 1, February 2014, pp. 31–36. c Indian Academy of Sciences A note on conjugacy classes of finite groups HEMANT KALRA and DEEPAK GUMBER School of Mathematics and Computer Applications, Thapar University, Patiala 147 004, India E-mail: [email protected]; [email protected] MS received 29 August 2012; revised 27 November 2012 Abstract. Let G be a finite group and let xG denote the conjugacy class of an element x of G. We classify all finite groups G in the following three cases: (i) Each non-trivial conjugacy class of G together with the identity element 1 is a subgroup of G, (ii) union of any two distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G, and (iii) union of any three distinct non-trivial conjugacy classes of G together with 1 is a subgroup of G. Keywords. Conjugacy class; rational group. 2000 Mathematics Subject Classification. 20E45, 20D20. 1. Introduction Throughout this paper, unless or otherwise stated, G denotes a finite group. The influence of arithmetic structure of conjugacy classes of G, like conjugacy class sizes, the number of conjugacy classes or the number of conjugacy class sizes, on the structure of G is an extensively studied question in group theory. Many authors have studied the influence of some other kind of behaviours of conjugacy classes on the structure of the group. For example, Dade and Yadav [2] have classified all finite groups in which the product of any two non-inverse conjugacy classes is again a conjugacy class.
    [Show full text]
  • If There Is F ∈ G Such That − G = F · H · F 1
    LECTURE 5 Conjugacy PAVEL RU˚ZIˇ CKAˇ Abstract. We study the conjugacy relation on a group. We prove that the conjugacy classes form a partition of the group. We show that a subgroup is normal if and only if it is a union of conjugacy classes. We will study some examples. we prove that permutations in a symmetric group are conjugated if and only if they have the same type. Finally we study sets of generators of a group. Understanding some sets of generators of an alternating group will help us to prove that all standard positions of the 15 puzzle corresponding to even permutations are solvable. 5.1. Conjugacy. Elements g, h of a group G are said to be conjugate (which we denote by g ∼ h) if there is f ∈ G such that − g = f · h · f 1. In this case we say the g is conjugate to h by f. Clearly g is conjugate to g by the unit of G and if g is conjugate to h by f then h is conjugate to g by f −1. It follows that the relation ∼ is reflexive and symmetric. If g is conjugate with h by f and h is conjugate with k by e, then g is conjugate with k by f · e, indeed, (f · e) · k · (f · e)−1 = f · e · k · e−1 · f −1 = f · h · f −1 = g. Thus we have the transitivity of ∼. We conclude that the conjugacy form an equivalence relation on G. The blocks of ∼ are called the conjugacy classe (of conjugacy blocks) of G.
    [Show full text]